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Physica B 350 (2004) 262–264

*Corresp

6421-282-89

E-mail a

(S. Hosoka

0921-4526/$

doi:10.1016

Microscopic dynamics in trivalent liquid Ga

S. Hosokawaa,*, W.-C. Pilgrima, H. Sinnb, E.E. Alpb

a Institut f .ur Physikalische-, Kern- und Makromolekulare Chemie, Philipps Universit .at Marburg, D-35032 Marburg, GermanybSRI-CAT, Advanced Photon Source, Argonne National Laboratory, Argonne, IL 60439, USA

Abstract

The dynamic structure factor SðQ;oÞ of liquid Ga was measured at 100�C using high-resolution inelastic X-ray

scattering. The obtained spectra clearly demonstrate the existence of longitudinal propagating modes at small Q: TheQ-o relation of the excitations shows a so-called positive dispersion of about 15%, which is much smaller than findings

from an inelastic neutron-scattering experiment carried out at higher temperature. The spectra are well reproduced by

an analysis using memory function formalism with two viscous decay channels.

r 2004 Elsevier B.V. All rights reserved.

PACS: 63.50.+x; 61.10.Eq; 61.25.Mv

Keywords: Liquid metal; Inelastic X-ray scattering; Phonon dispersion

1. Introduction

Ga is a fascinating metal, which possesses avariety of morphological crystalline forms includ-ing both, covalent and metallic properties. LiquidGa has an extremely wide liquid range from 29.8–2200�C, and can easily be supercooled down toB–120�C. It is expected that both covalent andmetallic characters may coexist in the liquid state.Thus, the flexible variation of the interatomiccorrelations and the local liquid structures withtemperature may induce such an unusually wideliquid range.The microscopic dynamics of liquid Ga was

investigated by inelastic neutron scattering (INS)

onding author. Tel.: +49-6421-282-2396; fax: +49-

16.

ddress: hosokawa@mailer.uni-marburg.de

wa).

- see front matter r 2004 Elsevier B.V. All rights reserve

/j.physb.2004.04.040

[1,2]. Well-defined collective excitations were notfound near the melting point, but at highertemperature of 700�C. It should, however, benoted that the frequencies of the short wavelengthmodes increase noticeably faster (B30%) thanpredicted by classical hydrodynamics, in contrastto results from a first-principles molecular dy-namics (MD) simulation [3]. In contrast to the INSdata, a recent inelastic X-ray scattering (IXS) [4]demonstrated the existence of acoustic-like modesnear the melting point. In this paper, we reportSðQ;oÞ results of liquid Ga at 100�C measuredover a wide Q range of 3.0–28.0 nm�1.

2. Experimental

The present experiments were carried out usinga horizontal IXS spectrometer at 3-ID-C/APS [5].

d.

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0.7

0.6

0.5

0.4

)/S

(Q)

liquid Ga 100 °C

Q (nm-1) =

25.0

19.0

S. Hosokawa et al. / Physica B 350 (2004) 262–264 263

There, a highly resolved monochromatized X-raybeam was achieved employing two nested Sichannel-cut crystals, and an (18 6 0) backscatteringreflection of a two-dimensionally curved Si analy-zer was used for the energy analysis of thescattered radiation. The overall energy resolutionof the spectrometer was 1.9meV FWHM, and theQ resolution was 70.45 nm�1. The Ga sample wascontained in a thin-walled (0.25mm) single-crystalsapphire cell [6], which was placed in a vesselequipped with continuous Be windows [7] capableof covering scattering angles between 0� and 25�.

0.3

0.2

0.1

0.0

S(Q

,ω

-40 -20 0 20 40ω (meV)

13.0

8.0

3.0

5.0

Fig. 1. SðQ;oÞ normalized to SðQÞ at 100�C at selected Q

values (circles with error bars). Solid lines represent the best fits

of a model using a memory function analysis convoluted with

the resolution function (dashed line).

3. Results and discussion

Fig. 1 shows the inelastic scattering spectranormalized to the respective integral intensity atselected Q values. This representation is almostidentical to SðQ;oÞ=SðQÞ; if the resolution broad-ening and the thermal balance factor is neglected.At low Q values, inelastic excitations are clearlyseen as peaks or shoulders at both the sides of thecentral line. With increasing Q; the energy posi-tions of the modes increase, which clearly demon-strates that the excitations originate frompropagating modes. Their width broadens withincreasing Q; and eventually appears to be highlydamped at QB13–19 nm�1.A generalized Langevin formalism employing a

suitable memory function was used to analyze thespectra. For the memory function, we used a well-known approximation containing two exponentialdecay channels for viscous relaxation and anadditional exponential for thermal relaxation. Thisapproach has proven to be useful in describingresults of computer simulation studies on simpleliquids [8], and also of experimental IXS data onliquid alkali metals [9]. For each Q value, themodel SðQ;oÞ convoluted with the experimentalresolution function was fitted to the data. Detailsof this method are described elsewhere [8,9].Solid lines in Fig. 1 represent best fits of thismodel using the memory function analysis con-voluted with the resolution function. As can beseen, the model function reproduces the experi-mental data very well.

The resolution-deconvoluted lineshape ofSðQ;oÞ could be obtained using the fittingparameters in the model function. Then, themaximum frequency oj of the correspondinglongitudinal current correlation function JlðQ;oÞ ¼ðo2=Q2ÞSðQ;oÞ was used to determine the velocityof sound at finite Q for liquid Ga. Closed circles inFig. 2 indicate the sound velocity obtained fromcðQÞ ¼ ojðQÞ=Q at various Q values. The presentresults are in good agreement with the previousfindings by Scopigno et al. [4] at 42�C shown by adot–dashed line.The arrow indicates the hydrodynamic limit at

Q-0; i.e., the adiabatic sound velocity of2854ms�1 at 100�C [10]. As seen in the figure,the cðQÞ values in the low Q region are larger than

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5000

4000

3000

2000

1000

0

c (m

/s)

302520151050

Q (nm-1)

liquid Ga 100 °C

Fig. 2. Sound velocity obtained from cðQÞ ¼ ojðQÞ=Q at 100�C

(closed circles). The arrow shows the adiabatic sound velocity

of 2854ms�1 [10]. Triangles and inverted triangles indicate

lower and upper limits, respectively, computed from the fitting

parameters. Also given are literature IXS results for cðQÞ (dot–dashed line) and the lower limit (solid line) at 42�C [4].

S. Hosokawa et al. / Physica B 350 (2004) 262–264264

those predicted by hydrodynamics by about 15%.This so-called positive dispersion was found earlierin several liquid metals, and as already mentionedabove—as a larger effect (B30%)—in the high-temperature INS data of Ga by Bermejo et al. [2].This is a surprising result, since the positivedispersion would rather decrease with increasingtemperature. Additionally, results of a first-princi-ples MD simulation by Holender et al. [3] showedalmost no positive dispersion even at hightemperatures of 429�C and 709�C, which ishowever in good agreement with the present IXSresults.The positive dispersion is explained within the

framework of generalized hydrodynamics [11],where it can be shown that frequency-dependentupper and lower limits exist for the propagationspeed of longitudinal collective modes. The trian-gles in Fig. 2 indicate the generalized isothermalsound velocity (lower limit), and the invertedtriangles the high-frequency-limit values, both ofwhich can also be calculated from the fittingparameters of the present analysis. Even at the

lowest Q value measured, cðQÞ deviates from thehydrodynamic prediction. With increasing Q; cðQÞalmost reaches the high-frequency limit at aboutQ=5nm�1, and approaches the lower limit closeto the SðQÞ maximum. This behaviour is wellknown from several liquid metals.

4. Conclusion

SðQ;oÞ of liquid Ga was measured at 100�Cusing high-resolution IXS. The obtained spectraclearly demonstrate the existence of longitudinalpropagating modes at small Q: The Q–o relationof the excitations shows a so-called positive

dispersion of about 15%, which is much smallerthan the result of a previous INS experimentmeasured at higher temperature. The spectra werewell reproduced by an analysis using memoryfunction formalism with two viscous decaychannels.

Acknowledgements

The authors acknowledge Prof. S. Takeda andDr. Y. Kawakita of Kyushu University for theirhelp in the experiment. We are grateful for thesupport from US DOE BES Materials Science,Contract No. W-31-109-ENG-38.

References

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